In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)

Verified step by step guidance

1. Understand that the problem asks to find the union of two intervals: \((-\infty, 5)\) and \([1, 8)\), which means combining all values that lie in either interval.
2. Recall that \((-\infty, 5)\) includes all real numbers less than 5, but does not include 5 itself, while \([1, 8)\) includes all real numbers from 1 up to but not including 8, including 1.
3. Visualize the intervals on a number line: the first interval covers everything to the left of 5, and the second interval covers from 1 to just before 8.
4. Notice that these intervals overlap between 1 and 5, so when combined, the union will cover from \(-\infty\) up to 8, but we need to check the endpoints carefully.
5. Express the union by combining the intervals into one continuous interval: since \((-\infty, 5)\) covers everything less than 5 and \([1, 8)\) covers from 1 to 8, the union is \((-\infty, 8)\), where 8 is not included.

Key Concepts

Interval Notation

Interval notation is a way to represent sets of real numbers using parentheses and brackets. Parentheses, like (a, b), indicate that endpoints are not included, while brackets, like [a, b], mean the endpoints are included. Understanding this helps interpret the given sets correctly.

Union of Sets

The union of two sets combines all elements from both sets without duplication. In interval notation, the union symbol (⋃) merges intervals, so any number in either interval is included in the final set. This concept is essential to combine (-∞, 5) and [1, 8) properly.

Graphing Intervals on the Number Line

Graphing intervals involves shading regions on the number line to represent all numbers within the interval. Open circles denote excluded endpoints, and closed circles denote included endpoints. Visualizing intervals helps understand their union and identify the combined set.